On the resolution of the Diophantine equation U_n + U_m = x^q

2022/02/24 by Pritam Kumar Bhoi, Bhoi, P. K., S. S. Rout +3
Mathematics · #11B39 #11D61 #11J86 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2202.11934

openalex publication_date 2022/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose that (U_n)_{n ≥ 0} is a binary recurrence sequence and has a dominant root α with α>1 and the discriminant D is square-free. In this paper, we study the Diophantine equation U_n + U_m = x^q in integers n ≥ m ≥ 0, x ≥ 2, and q ≥ 2. Firstly, we show that there are only finitely many of them for a fixed x using linear forms in logarithms. Secondly, we show that there are only finitely many solutions in (n, m, x, q) with q, x≥ 2 under the assumption of the \em abc-conjecture. To prove this, we use several classical results like Schmidt subspace theorem, a fundamental theorem on linear equations in S-units and Siegel's theorem concerning the finiteness of the number of solutions of a hyperelliptic equation.

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